DESIGN OF THE SAMPLE QUESTION PAPERS MATHEMATICS CLASS X Time : 3 Hours Max. Marks : 100 The weightage or the distribution of marks over different dimensions of the question papers shall be as follows : 1. Weightage to Learning Outcomes S.No. Learning Outcomes Marks 1. Knowledge 31 2. Understanding 45 3. Application 12 4. Skill 12 2. Weightage to content/subject Unit S.No. Learning Outcomes Marks 1. Algebra 26 2. Commercial Mathematics 12 3. Mensuration 10 4. Trigonometry 10 5. Geometry 22 6. Statistics 12 7. Coordinate Geometry 8 3. Weightage to form of questions Total : 100 S. No. Form of Question Marks for each question Number of questions Total Marks 1. SA II 3 10 30 2. SA I 4 10 40 3. LA 6 05 30 4. The expected length of answer under different forms of questions and expected time would be as follows : S. No. Form of Questions No. of credit points Approx. Time 1. Short answer type (SA I) Upto 4 Credit Points 3 5 minutes 2. Short answer type (SA II) Upto 6 Credit Points 5 7 minutes 3. Long answer type (LA) Upto 8 Credit Points 8 10 minutes
These ranges of steps and time requirements for the answers are, however, suggestive. In practice, actual number of steps and time needed may vary. As the total time is calculated on the basis of the number of questions required to be answered and the length of their anticipated answers, it would, therefore, be advisable for the candidates to budget their time properly by cutting out the superfluous lengths and be within the expected limits. 5. Scheme of Options All questions are compulsory i.e. there is no overall choice in the question paper. However, internal choices have been provided in two questions of 3 marks each, two questions of 4 marks each and two questions of 6 marks each. These choices have been given from within the same topic and in questions which test higher mental abilities of students. 6. Weightage to difficulty level of questions S. No. Estimated Difficulty Level of Questions % of Marks 1. Easy 15% 2. Average 70% 3. Difficult 15% A question may vary in difficulty level from individual to individual. As such, the assessment in respect of each question will be made by the paper setter on the basis of general anticipation from the group as whole taking the examination. This provision is only to make the paper balanced in its weight, rather to determine the pattern of marking at any stage. Based on the above design, there are two separate sample papers along with their Blue Prints as well as questionwise analysis. For the examination of the Board, while the design of the question papers will remain same, blue prints based on this design may change. Note : Though weightages to content/subject units, objectives and forms of questions etc. have been clearly assigned, yet depending on the exigencies of the paper, these can vary to some extent in Board s examination.
Sample Question Paper - I Class X Subject : Mathematics Time : 3 Hours Max. Marks : 100 General Instructions 1. All questions are compulsory. 2. The question paper consists of 25 questions divided into three sections A,B and C. Section A contains 10 questions of 3 marks each, Section B is of 10 questions of 4 marks each and Sections C is of 5 questions of 6 marks each. 3. There is no overall choice. However, and internal choice has been provided in two questions of three marks each, two questions of four marks each and two questions of six marks each. 4. Write the serial number of the question before attempting it. 5. In question on construction, the drawing should be neat and exactly as per the given measurements. 6. Use of calculators is not permitted. However, you may ask for Mathematical tables. SECTION A Q1. Solve the following system of equations : 15x + 4y = 61 4x + 15y = 72 Q2. Reduce the following rational expression to its lowest terms : 2 3 x + 3x + 9 x 27 2 2 x 25 ( x + 3x 10)
Q3. PQ and RS are two parallel chords of a circle and the lines RP and SQ meet at O on producing (as shown in the given figure) Prove that OP=OQ Q4. A suit is available for Rs. 1500 cash or for Rs. 500 cash down payment followed by 3 monthly instalments of Rs. 345 each. Find the rate of interest charged under the instalment scheme. Q5. A loan has to be returned in two equal annual instalments. If the rate of interest is 16% per annum compounded annually and each instalment is of Rs. 1682, find the sum borrowed and the total interest paid. 2 Q6. If ( x 2) is a factor of x + ax + b and a + b = 1, find the values of a and b. Q7. Using quadratic formula, solve the following equation for x : abx 2 + ( b 2 ac) x bc = 0 The sum of the squares of two positive integers is 208. If the square of the larger number is 18 times the smaller, find the numbers. Q8. Which term of the A.P. 3,15,27,39. is 132 more than its 54 th term? Derive the formula for the first n terms of an A.P. whose first term is a and the common difference is d Q9. Find the sum of the following arithmetic progression 1+3+5+7+....+199 Q10. Show that a line drawn parallel to the parallel sides of a trapezium divides the non parallel sides proportionally.
SECTION B 1 2 4 Q.11. Solve for x : + =, x + 1 x + 2 x + 4 ( x 1, 2, 4) Q12. Find graphically, the vertices of the triangle formed by the x-axes and the lines 2x - y + 8 = 0 8x + 3y 24 = 0 Q13. Construct a triangle ABC in which BC = 13 cm, CA = 5 cm and AB = 12 cm. Draw its incircle and measure its radius. Q14. The total surface area of a right circular cylinder is 6512 cm 2, and the circumference of 22 its base is 88 cm. Find the volume of the cylinder (Use π = ) 7 Q15. Prove the identity : (1+Cotθ - Cosecθ) (1+tanθ + secθ) = 2. Without using trigonometric tables, evaluate : o o cos35 tan 27 tan 63 2 + 3tan 60 o o sin 55 sin 30 o o Q16. Show that the points (7,10), (-2,5) and (3,-4) are the vertices of an isosceles right triangle. Using distance formula, show that the points (-1,-1), (2,3) and (8,11) are collinear. Q17. Find the ratio in which the point (-3,p) divides the line segment joining the points (-5,-4) and (-2,3). Hence find the value of p. Q.18. Compute the missing frequencies f 1 and f 2 in the following data if the mean is 9 166 and the sum of observations is 52. 26 Classes 140-150 150-160 160-170 170-180 180-190 190-200 Frequency 5 f 1 20 f 2 6 2 = 52
Q19. An unbiased dice is tossed. (i) Write the sample space of the experiment (ii) Find the probability of getting a number greater than 4 (iii) Find the probability of getting a prime number. Q20. The pie chart (as shown in the figure) represents the amount spent on different sports by a sports club in a year. If the total money spent by the club on sports is Rs. 1,08,000/-, find the amount spent on each sport. SECTION C Q21. Prove that the angle subtended by an arc of a circle at its center is double the angle subtended by it at any point on the remaining part of the circle. Using the above result prove that the angle in a major segment is acute. Q22. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. Using the above, prove that the area of an equilateral triangle described on the side of a square is half the area of the equilateral triangle described on its diagonal. Q23. From the top of a tower 60m. high, the angles of depression of the top and bottom of a building whose base is in the same straight line with the base of the tower are observed to be 30 0 and 60 0 respectively. Find the height of the building. An aeroplane flying horizontally at a height of 1.5 km above the ground is observed at a certain point on earth to subtend an angle of 60 0. After 15 seconds, its angle of elevation is observed to be 30 0. Calculate the speed of the aeroplane in km/h.
Q24. A solid toy is in the form of a hemisphere surmounted by a right circular cone. If the height of the cone is 4 cm and diameter of the base is 6 cm calculate : (i) the volume of the toy (ii) surface area of the toy (Use π = 3.14) A bucket of height 8 cm. and made up of copper sheet is in the form of frustrum of a right circular cone with radii of its lower and upper ends as 3 cm and 9 cm respectively. Calculate : (i) the height of the cone of which the bucket is a part (ii) (iii) the volume of water which can be filled in the bucket. the area of copper sheet required to make the bucket (Leave the answer in terms of π ) Q25. Anil s total annual salary excluding HRA is Rs. 1,96,000. He contributes Rs. 5000 per month in his G.P.F. How much he should invest in N.S.C. to get maximum rebate?. After getting maximum rebate he wants to pay income tax in equal monthly instalments. Find the amount which should be deducted per month towards tax from his salary. Assume the following for calculating income tax : a) Standard deduction : 1/3 of the total income subject to maximum of Rs. 25000 in case the total annual income is more than Rs. 1,50,000 b) Rate of income tax : Slab Income tax i) Up to Rs. 50,000 No tax ii) From Rs. 50,001 to Rs. 60,000 10% of the amount exceeding Rs. 50,000 iii) From Rs. 60,001 to Rs. 1,50,000 Rs. 1000 + 20% of the amount exceeding Rs. 60,000 iv) Above Rs. 1,50,000 Rs. 19,000 + 30% of the amount exceeding Rs. 1,50,000 c) Rebate in income tax : i) 20% of the amount of saving subject to maximum Rs. 14,000/-, if taxable income is upto Rs. 1,50,000 ii) 15% of the amount of saving subject to a maximum of Rs. 10,500/- if taxable income is above Rs. 1,50,000 d) Surcharge 5% of the net income tax paid
Sample Question Paper Set II Class X Subject : Mathematics Time : 3 Hours Max. Marks : 100 General Instructions 1. All questions are compulsory. 2. The question paper consists of 25 questions divided into three sections A,B and C. Section A contains 10 questions of 3 marks each, Section B is of 10 questions of 4 marks each and Sections C is of 5 questions of 6 marks each. 3. There is no overall choice. However, and internal choice has been provided in two questions of three marks each, two questions of four marks each and two questions of six marks each. 4. Write the serial number of the question before attempting it. 5. In question on construction, the drawing should be neat and exactly as per the given measurements. 6. Use of calculators is not permitted. However, you may ask for Mathematical tables. SECTION A Q1. Solve the following system of equations graphically 5x - y = 7 x y = -1 Q2. Find the Arithmetic Progression whose third term is 16 and the seventh term exceeds its fifth term by 12. Q3. ABD is a triangle in which DAB = 90 0. AC is drawn perpendicular from A to DB. Prove that : AD 2 = BD x CD Q4. A loan of Rs. 48,800/- is to be paid back in three equal annual instalments. If the rate of interest is 25% per annum compounded annually, find the instalment.
Q5. A watch is available for Rs. 970 cash or Rs. 210 as cash down followed by three equal monthly instalments. If the rate of interest if 16% per annually, find the monthly instalment. Q6. Construct the pair of tangents drawn from a point, 5 cm away from the center of a circle of radius 2 cm. Measure the lengths of the tangents. Q7. A solid metallic cylinder of radius 14 cm and height 21 cm is melted and recast into 72 equal (small spheres. Find the radius of one such sphere. Q8. The rain water from a roof 22m x 20m drains into a conical vessel having diameter of base as 2m and height 3.5m. If the vessel is just full, find the rainfall (in cm.) The largest sphere is carved out of a cube of side 7 cm; find the volume of the sphere. Q9. The following table shows the marks secured by 100 students in an examination Marks 0-10 10-20 20-30 30-40 40-50 Number of students 15 20 35 20 10 Find the mean marks obtained by a student. Q10. A dice is thrown once. Find the probability of getting (i) a number greater than 3 (ii) a number less than 5 A bag contains 5 red balls, 8 white balls, 4 green balls and 7 black balls. A ball is drawn at random from the bag. Find the probability that it is (i) (ii) black not green SECTION B Q11. Solve for x and y 2 2 ( a b) x + ( a + b) y = a 2ab b 2 2 ( a + b)( x + y) = a + b
Q12. If ( x + 3)( x 2) is the G.C.D. of 2 f ( x) = ( x + 3)(2x 3x + a) 2 and g( x) = ( x 2)(3x + 10x b) find the value of a and b Q13. If 2x + 1 2x 1 A =, B = find 2x 1 2x + 1, A + B + A B A B A + B Q14. Solve for x : x x 1 + + 2 x x 3 = 4 10 3 ( x 2, x 4 ) Q15. A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. Find the usual speed of the train. Q16. AB is a diameter of a circle with center O and chord CD is equal to radius of the circle. AC and BD are produced to meet at P. Prove that CPD = 60 0 Q17. A circus tent is in the shape of a cylinder surmounted by a cone. The diameter of the cylindrical part is 24 m and its height is 11 m. If the vertex of the tent is 16m above the ground, find the area of canvas required to make the tent. Q18. Prove that tanθ cotθ + = 1+ secθ cos ecθ 1 cotθ 1 tanθ
Evaluate : sin39 cos51 0 0 0 0 0 0 0 + 2tan11 tan31 tan 45 tan59...tan 79 3 sin 2 0 2 0 ( 21 + sin 69 ) Q19. Find a point on the x-axis which is equidistant from the points (7,6) and (-3,4) Q20. Three consecutive vertices of a parallelogram ABCD are A(1,2) B(1,0) and C(4,0). Find the fourth vertex D. If A(4,-8), B(-9,7) and C(18,13) are the vertices of a triangle ABC, find the length of the median through A and coordinates of centroid of the triangle. SECTION C Q21. The number of hours spent by a school boy on various activities on a working day are given below : Activity Sleep 7 School 8 Homework 4 Play 3 Others 2 Number of Hours Present the above information by a pie-chart. Q22. A vertical tower is surmounted by a flagstaff of height h metres. At a point on the ground, the angles of elevation of the bottom an top of the flagstaff are α and β respectively. Prove that the height of the tower is : h tanα tan β tanα
If the angle of elevation of a cloud from a point h meters above a lake is α and the angle of depression of its reflection in the lake is β, prove that the distance of the cloud from the point of observation is 2hsecα tan β tanα Q23. If a line is drawn parallel to one side of a triangle, prove that the other two sides are divided in the same ratio. Using the above result, prove the following : The diagonals of a trapezium divide each other in the same ratio Q24. Prove that the sum of either pair of the opposite angles of a cyclic quadrilateral is 180 0. Using the above result, determine as under : ABCD is a cyclic trapezium with AD BC. If B = 70 0, determine the other three angles of the trapezium If two circles touch each other internally or externally, prove that the point of contact lies on the line joining their centers. Using the above result prove the following Two circles with centers O and O / and radii r 1 and r 2 touch each other externally at P. AB is a line through P intersecting the two circles at A & B respectively. Prove that OA OB /. Q25. Ramlal has a total annual income of Rs.1,60,000/-. He contributes Rs.4000 per month in his GPF and pays an annual LIC premium of Rs. 15,000. If he pays Rs. 250 per month for first 11 months, find his income tax liability for the last month. Use the following for calculating income tax :
a) Standard Deduction If the salary exceeds Rs. 1,50,000 but does not exceed Rs. 3,00,000, the standard deduction is Rs. 25,000 b) Rates of Income Tax i) Upto Rs. 50,000 No tax ii) Rs. 50,001 to Rs. 60,000 10% of the amount exceeding Rs. 50,000 iii) Rs. 60,001 to Rs. 1,50,000 Rs. 1000 + 20% of the amount exceeding Rs. 60,000. c) Rebate on Savings 20% of the total savings if the taxable income is upto 1,50,000 subject to a maximum of Rs. 14,000 d) Surcharge 5% of the tax payable.